Shakespeare’s play is called A Midsummer Night’s Dream.
So is A Midsummer’s Night’s Dream correct English?
If not, what would be the correct English?
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Shakespeare's title parses as "A (Midsummer Night)'s dream".
That is, a dream belonging to a midsummer night: a dream, or kind of dream, one has (or had) on a night in the middle of summer. This is contrasted with, say, the kind of dream you'd have on a rainy Tuesday night in February. In other words, though "Midsummer" and "Night" are technically two separate words, they're being treated as an indivisible unit.
Your alternative, though logical, loses somewhat of the original's evocative quality by breaking up this unit. You're saying "a dream that belongs to a night that belongs to a midsummer". Now, instead of the "kind of dream one has on a midsummer night", you're describing a dream you had on some night in the middle of some summer.
I'll also caution you that, this being a famous work, if you were actually to say or write the title with two possessives, you might put off a finickier audience (I guarantee someone would have the urge to correct you), which at best would be a distraction, and at worst might create some prejudices.
That said, if Shakespeare's work is just an example, and you're interested in the use of possessives in a more general way, your phrasing is clear and anyone hearing it would understand you. The distinction is really whether (Midsummer Night) is conceptually indivisible or the two things can operate independently (some night of some summer).
“A Midsummer’s Night’s Dream” is quite grammatical, and means a dream dreamed during one of the nights of a mid-summer. It just is not the standard title of Shakespeare’s play. (Actually, neither the Q1 title page, nor in F1 the TOC or the play’s first page or the running header uses any apostrophe at all in the title, but “Midsommer” never has a terminal s and “Night” always does.)
I frequently hear the title garbled as “A Midsummer’s Night-Dream.” (And of course hardly anyone manages to do the apostrophes correctly in Love’s Labor’s Lost.)
When examined as phrases, both are compound noun phrases.
In case of "A Midsummer Night's Dream"
Midsummer Night is a compound noun adjunct phrase with Night as the head-noun and Midsummer as the adjunct.
So if you were to parse the phrase out:
((((Midsummer:adjunct noun) Night:head noun)): noun adjunct)'s: possessive) (Dream: head-noun)
In case of "A Midsummer's Night's Dream" using a possessive in the inner phrase makes the word Midsummer the head-noun instead.
(((((Midsummer:head-noun)'s: possessive) Night:noun adjunct)): noun adjunct)'s: possessive) (Dream: head-noun)
These subtleties are further, and nicely, discussed in A noun adjunct / the possessive case